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Theorem fmf 22526
Description: Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fmf ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Proof of Theorem fmf
Dummy variables 𝑓 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7164 . . . 4 (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))) ∈ V
2 eqid 2820 . . . 4 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
31, 2fnmpti 6465 . . 3 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) Fn (fBas‘𝑌)
4 df-fm 22519 . . . . . 6 FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))))
54a1i 11 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))))))
6 dmeq 5746 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
76adantl 484 . . . . . . . 8 ((𝑥 = 𝑋𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
8 fdm 6496 . . . . . . . . 9 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
983ad2ant3 1131 . . . . . . . 8 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → dom 𝐹 = 𝑌)
107, 9sylan9eqr 2877 . . . . . . 7 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → dom 𝑓 = 𝑌)
1110fveq2d 6648 . . . . . 6 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (fBas‘dom 𝑓) = (fBas‘𝑌))
12 id 22 . . . . . . . 8 (𝑥 = 𝑋𝑥 = 𝑋)
13 imaeq1 5898 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1413mpteq2dv 5136 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑦𝑏 ↦ (𝑓𝑦)) = (𝑦𝑏 ↦ (𝐹𝑦)))
1514rneqd 5782 . . . . . . . 8 (𝑓 = 𝐹 → ran (𝑦𝑏 ↦ (𝑓𝑦)) = ran (𝑦𝑏 ↦ (𝐹𝑦)))
1612, 15oveqan12d 7150 . . . . . . 7 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
1716adantl 484 . . . . . 6 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
1811, 17mpteq12dv 5125 . . . . 5 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
19 elex 3491 . . . . . 6 (𝑋𝐴𝑋 ∈ V)
20193ad2ant1 1129 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → 𝑋 ∈ V)
21 fex2 7614 . . . . . 6 ((𝐹:𝑌𝑋𝑌𝐵𝑋𝐴) → 𝐹 ∈ V)
22213com13 1120 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → 𝐹 ∈ V)
23 fvex 6657 . . . . . . 7 (fBas‘𝑌) ∈ V
2423mptex 6960 . . . . . 6 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V
2524a1i 11 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V)
265, 18, 20, 22, 25ovmpod 7277 . . . 4 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
2726fneq1d 6420 . . 3 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ↔ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) Fn (fBas‘𝑌)))
283, 27mpbiri 260 . 2 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
29 simpl1 1187 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐴)
30 simpr 487 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
31 simpl3 1189 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
32 fmfil 22525 . . . 4 ((𝑋𝐴𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
3329, 30, 31, 32syl3anc 1367 . . 3 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
3433ralrimiva 3169 . 2 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
35 ffnfv 6856 . 2 ((𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋) ↔ ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ∧ ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋)))
3628, 34, 35sylanbrc 585 1 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473  cmpt 5120  dom cdm 5529  ran crn 5530  cima 5532   Fn wfn 6324  wf 6325  cfv 6329  (class class class)co 7131  cmpo 7133  fBascfbas 20506  filGencfg 20507  Filcfil 22426   FilMap cfm 22514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7134  df-oprab 7135  df-mpo 7136  df-fbas 20515  df-fg 20516  df-fil 22427  df-fm 22519
This theorem is referenced by:  rnelfm  22534
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